Abstract
Some qualitative properties of positive smooth solutions to a generalized nonlinear parabolic equation involving f-Laplacian (L-f)
(L-f - q(x, t)-partial derivative/dt) w(x, t) = G(w(x, t)),
are discussed on M(f)x(-infinity, +infinity), where M-f is a complete smooth metric measure space (with or without boundary), the potential function q(x, t) is smooth at least C-1 in xand C-0 in t, and G( w(x, t)) is a nonlinear smooth sourcing term. Local and global type space only (elliptic type) gradient estimates are established for this equation under the condition that Bakry-Emery Ricci curvature tensor is bounded from below. As an exploitation of the gradient estimates so derived we obtain a parabolic Harnack inequality and some Liouville type theorems for bounded ancient and eternal solutions. The approach adopted in this paper provides a unified treatment of a large class of nonlinear source terms. To demonstrate this further, cases where G(w) = aw(rho), a is an element of R, rho is an element of(-infinity, 0] boolean OR [1, +infinity), G(w) = aw vertical bar logw vertical bar(gamma), a is an element of R, a not equal 0, gamma > 1 and G(w) = aw(logw)gamma a not equal 0, gamma= 1 are considered as specific examples. We further show that all the obtained results also hold on weighted manifolds with compact boundary under some lower boundedness assumptions on mean curvature of the boundary and Bakry-Emery Ricci curvature tensor. (C) 2022 Elsevier B.V. All rights reserved.